Multisection of series

2016 ◽  
Vol 100 (549) ◽  
pp. 460-470
Author(s):  
Raymond A. Beauregard ◽  
Vladimir A. Dobrushkin
Keyword(s):  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Binomial Coefficients ◽  
Discrete Mathematics ◽  
Mathematical Tool ◽  
New Birth ◽  
Leonhard Euler ◽  
Finite Sums ◽  
Symbolic Methods ◽  
Infinite Sums

In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients.While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals ON GENERALIGATIONS OF PARTITION FUNCTIONS

2015 ◽  
Vol 3 (10) ◽  
pp. 1-29
Author(s):  
Nil Ratan Bhattacharjee ◽  
Sabuj Das
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Partition Functions ◽  
Auxiliary Functions ◽  
Numerical Example ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook ◽  
Leonhard Euler

In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for   X(n),Y(n) . In 2014, Sabuj developed the generating functions for .  In 2005, George E. Andrews found the generating functions for    In 1916, Ramanujan showed the generating functions for  ,  ,   and  . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan’s Lost Notebook. In 1967, George E. Andrews defined the generating functions for P1r (n) and P2r (n). In this article these generating functions are discussed elaborately. This article shows how to prove the theorem P2r (n) = P3r (n) with a numerical example when n = 9 and r = 2. In 1995, Fokkink, Fokkink and Wang defined the   in terms of , where   is the smallest part of partition . In 2013, Andrews, Garvan and Liang extended the FFW-function and defined the generating function for FFW (z, n) in differnt way.

scholarly journals A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials

2020 ◽  
pp. 42-42
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Recurrence Formula ◽  
Binomial Coefficients ◽  
Combinatorial Identity ◽  
Lucas Numbers ◽  
Fundamental Properties ◽  
Special Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers are given.

XIX.—Asymptotic enumeration of connected graphs

1970 ◽  
Vol 68 (4) ◽  
pp. 298-308
Author(s):  
E. M. Wright
Keyword(s):  
Asymptotic Expansion ◽  
Generating Functions ◽  
Binomial Coefficients ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Connected Graphs

SynopsisThe number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

scholarly journals A Note on the Difference of Powers and Falling Powers

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Taoufik Sabar
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
Special Series ◽  
Induction Principle ◽  
Combinatorial Sums ◽  
The Difference ◽  
Binomial Identities

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

scholarly journals New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

2021 ◽  
pp. 15-15
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Discrete Mathematics ◽  
New Class ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Derivative Formulas ◽  
Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

From golden-ratio equalities to Fibonacci and Lucas identities

2013 ◽  
Vol 97 (539) ◽  
pp. 234-241
Author(s):  
Martin Griffiths
Keyword(s):  
High School Students ◽  
Generating Functions ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
School Students ◽  
Simple Method ◽  
Matrix Methods ◽  
Binet’S Formula

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

scholarly journals The location of the first maximum in the first sojourn of a Dyck path

2008 ◽  
Vol Vol. 10 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Random Walks ◽  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Dyck Path ◽  
Dyck Paths ◽  
Asymptotic Expressions ◽  
International Audience

Analysis of Algorithms International audience For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.

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